I don't know much about DNG files.
Lightroom resize to Fit Width and Height may be changing the aspect ratio. Let's test. 3840/6000=0.64 and 2160/3376=0.6398. Thus your results from Lightroom show that the resulting image is preserving the aspect ratio. ImageMagick will also preserve the aspect ratio, unless you use the flag "!", which you did not.
Running
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convert -size 6000x3376 xc:white -resize "3840x2160>" -format "%wx%h" info:
3839x2160
This is only 1 pixel different from 3840x2160
So something is wrong with your convert command or something is odd in your DNG file or you misreported the ImageMagick resulting file size.
Since DNG is a photoshop proprietary format and is a type of raw format, the image may be different size than you specified due to unseen borders or virtual canvas. Lightroom may only report the size you have specified. But typically for raw formats you have to tell it the size. This may or may not be needed for DNG if that is known to photoshop and not to the DNG reader due to it being stored in photoshop proprietary tags. Again I know little about DNG and one of the DNG users may be able to shed more light.
What is your IM version and platform/OS?
Please always provide that. Also it would be helpful if you post your DNG file to some free hosting service that won't change the format or zip it first and put the URL here. Then we can review the file better and test with it.
EDIT:
According to the ImageMagick Formats page:
DNG R Digital Negative Requires an explicit image format otherwise the image is interpreted as a TIFF image (e.g. dng:image.dng).
So try
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convert dng:image.dng -resize "3840x2160>" image.jpg